\(\int e^{-2 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [789]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\left (c-\frac {c}{a^2 x^2}\right )^p x+\frac {2 (1-p) \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-p,\frac {3}{2},\frac {1}{a^2 x^2}\right )}{a^2 x}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a^2 x^2}\right )}{a p} \] Output:

-(c-c/a^2/x^2)^p*x+2*(1-p)*(c-c/a^2/x^2)^p*hypergeom([1/2, 1-p],[3/2],1/a^ 
2/x^2)/a^2/((1-1/a^2/x^2)^p)/x+(c-c/a^2/x^2)^p*hypergeom([1, p],[p+1],1-1/ 
a^2/x^2)/a/p
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} \left (-\left (-1+a^2 x^2\right )^2\right )^{-p} \left (-2 (-1+a x)^p \left (1-a^2 x^2\right )^p \operatorname {AppellF1}(1-2 p,-p,1-p,2-2 p,a x,-a x)+(1-a x)^p \left (-1+a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,a^2 x^2\right )\right )}{-1+2 p} \] Input:

Integrate[(c - c/(a^2*x^2))^p/E^(2*ArcTanh[a*x]),x]
 

Output:

((c - c/(a^2*x^2))^p*x*(-2*(-1 + a*x)^p*(1 - a^2*x^2)^p*AppellF1[1 - 2*p, 
-p, 1 - p, 2 - 2*p, a*x, -(a*x)] + (1 - a*x)^p*(-1 + a^2*x^2)^p*Hypergeome 
tric2F1[1/2 - p, -p, 3/2 - p, a^2*x^2]))/((-1 + 2*p)*(1 - a*x)^p*(-(-1 + a 
^2*x^2)^2)^p)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6709, 570, 559, 27, 557, 278}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx\)

\(\Big \downarrow \) 6709

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int \frac {x^{-2 p} \left (1-a^2 x^2\right )^{p+1}}{(a x+1)^2}dx\)

\(\Big \downarrow \) 570

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int x^{-2 p} (1-a x)^2 \left (1-a^2 x^2\right )^{p-1}dx\)

\(\Big \downarrow \) 559

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (x^{1-2 p} \left (-\left (1-a^2 x^2\right )^p\right )-\frac {\int -2 a^2 x^{-2 p} (-p-a x+1) \left (1-a^2 x^2\right )^{p-1}dx}{a^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \int x^{-2 p} (-p-a x+1) \left (1-a^2 x^2\right )^{p-1}dx-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\)

\(\Big \downarrow \) 557

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \left ((1-p) \int x^{-2 p} \left (1-a^2 x^2\right )^{p-1}dx-a \int x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}dx\right )-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\)

\(\Big \downarrow \) 278

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \left (\frac {(1-p) x^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),1-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a x^{2-2 p} \operatorname {Hypergeometric2F1}\left (1-p,1-p,2-p,a^2 x^2\right )}{2 (1-p)}\right )-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\)

Input:

Int[(c - c/(a^2*x^2))^p/E^(2*ArcTanh[a*x]),x]
 

Output:

((c - c/(a^2*x^2))^p*x^(2*p)*(-(x^(1 - 2*p)*(1 - a^2*x^2)^p) + 2*(((1 - p) 
*x^(1 - 2*p)*Hypergeometric2F1[(1 - 2*p)/2, 1 - p, (3 - 2*p)/2, a^2*x^2])/ 
(1 - 2*p) - (a*x^(2 - 2*p)*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2] 
)/(2*(1 - p)))))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 6709

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p)   Int[u*((1 + a*x)^n/(x^ 
(2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[c, 0]
 

Maple [F]

\[\int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}d x\]

Input:

int((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

int((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x)
 

Fricas [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
 

Output:

integral(-(a*x - 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a*x + 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.42 (sec) , antiderivative size = 699, normalized size of antiderivative = 6.24 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\text {Too large to display} \] Input:

integrate((c-c/a**2/x**2)**p/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

-a*Piecewise((0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(-1 + 
1/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 + c**p*p*x**(2 - 2*p)*e 
xp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2 
)/(2*a**(2*p)*gamma(2 - p)*gamma(p + 1)) + a**(3 - 2*p)*c**p*p*x**(3 - 2*p 
)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), 
a**2*x**2)/(2*a**2*gamma(p - 1/2)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0 
**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(1 - 1/(a**2*x**2))/( 
2*a**2) - 0**p*atanh(1/(a*x))/a**2 + c**p*p*x**(2 - 2*p)*exp(I*pi*p)*gamma 
(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*a**(2*p)*ga 
mma(2 - p)*gamma(p + 1)) + a**(3 - 2*p)*c**p*p*x**(3 - 2*p)*exp(I*pi*p)*ga 
mma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*a* 
*2*gamma(p - 1/2)*gamma(p + 1)), True)) + Piecewise((0**p*log(a**2*x**2 - 
1)/(2*a) + 0**p*acoth(a*x)/a - a**(1 - 2*p)*c**p*p*x**(2 - 2*p)*exp(I*pi*p 
)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamm 
a(2 - p)*gamma(p + 1)) - a**(1 - 2*p)*c**p*p*x**(1 - 2*p)*exp(I*pi*p)*gamm 
a(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*a*ga 
mma(p + 1/2)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1) 
/(2*a) + 0**p*atanh(a*x)/a - a**(1 - 2*p)*c**p*p*x**(2 - 2*p)*exp(I*pi*p)* 
gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma( 
2 - p)*gamma(p + 1)) - a**(1 - 2*p)*c**p*p*x**(1 - 2*p)*exp(I*pi*p)*gam...
 

Maxima [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
 

Output:

-integrate((a^2*x^2 - 1)*(c - c/(a^2*x^2))^p/(a*x + 1)^2, x)
 

Giac [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

integrate(-(a^2*x^2 - 1)*(c - c/(a^2*x^2))^p/(a*x + 1)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:

int(-((c - c/(a^2*x^2))^p*(a^2*x^2 - 1))/(a*x + 1)^2,x)
 

Output:

-int(((c - c/(a^2*x^2))^p*(a^2*x^2 - 1))/(a*x + 1)^2, x)
 

Reduce [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a x +x^{2 p}}d x -\left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p} x}{x^{2 p} a x +x^{2 p}}d x \right ) a}{a^{2 p}} \] Input:

int((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(int((a**2*c*x**2 - c)**p/(x**(2*p)*a*x + x**(2*p)),x) - int(((a**2*c*x**2 
 - c)**p*x)/(x**(2*p)*a*x + x**(2*p)),x)*a)/a**(2*p)